The present disclosure relates generally to regression analysis techniques and more particularly to regression analysis techniques using exponential random variables.
In statistics, regression analysis provides a statistical process for estimating the relationships among variables. These relationships can be used to establish techniques for modeling and analyzing several variables at once. The
focus of such analysis is based on the relationships existing between a dependent variable and one or a plurality of independent variables. Therefore, regression analysis is useful in understanding how values of the dependent variables change when any one of the independent variables is changed especially when other independent variables are held at a constant.
Embedding can be also used to aid regression analysis. In mathematics, an embedding is where a mathematical structure is contained within another group that is in turn a subgroup. An embedding can provide a one-to-one function that is a homeomorphism onto its image. An “oblivious subspace embedding” (OSE) is a type of embedding that provides distribution over matrices S such that for any low-dimensional subspace V, with high probability over the choice of S, ∥Sx∥_2 approximately equals ∥x∥_2 (up to 1+eps multiplicative error) for all x in V simultaneously.
Oblivious subspace embeddings have proven to be an essential ingredient for quickly and approximately solving numerical linear algebra problems such as used in regression analysis. Prior art provides that such embeddings could be used to approximately solve least squares regression and low rank approximation time. OSE can also be used for speeding up algorithms for several numerical linear algebra problems. Problems that benefit from OSE's may include approximate least squares regression, low-rank approximation, approximating leverage scores, and constructing good preconditioners. A precondition can be defined as a condition or predicate that must always be true. In computing environments, a precondition must be true prior to the execution of some section or all areas of the code or before an operation in a formal specification. Traditionally, if a precondition is violated, the effect of the calculation in statistical data or execution of the code in computing environments becomes undefined and thus may or may not carry out its intended work.